A short article designed to provide an introduction to complex variables, studying the effect of assuming differentiability of functions defined on complex
numbers. The effect is markedly different than for real functions; these functions are much more rigidly constrained, and in particular it is possible to make very definite comments about their global behavior, convergence, and so on. This area includes Riemann surfaces, which look locally like the complex plane but aren't the same space. Complex-variable techniques have great use in applied areas (including electromagnetics, for example). History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus.